Machine learning model for image recognition used in autonomous vehicles

ABSTRACT

An information processing device includes a first determination section and a first output section. First output data is obtained from a trained model, which has being trained beforehand, by input of first input data to the trained model. Second output data is obtained from the trained model by input of second input data to the trained model, in which second input data a perturbation of a specified perturbation amount is applied to the first input data. The first determination section makes a determination as to whether a change amount of the second output data relative to the first output data is not more than a pre-specified threshold. The first output section outputs information representing the first input data and a perturbation amount for which the change amount is determined by the first determination section to be not more than the threshold.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based on and claims priority under 35 USC § 119 fromJapanese Patent Application No. 2020-076792 filed on Apr. 23, 2020, thedisclosure of which is incorporated by reference herein.

TECHNICAL FIELD

The present disclosure relates to an information processing system.

RELATED ART

Xiang, Tran and Johnson, Reachable Set Computation and SafetyVerification for Neural Networks with ReLU Activations (arXiv2017)(Non-Patent Document 1) discloses a technology that employs a neuralnetwork to precisely identify ranges of output data of a trained model,which has been trained beforehand, that correspond with input data tothe trained model.

In the technology recited in Non-Patent Document 1, computationalcomplexity is very large, which is not practical. Therefore, there isscope for improvement in regard to reducing the computationalcomplexity. Furthermore, the technology recited in Non-Patent Document 1does not evaluate the reliability of output data obtained by inputtingdata to the trained model.

SUMMARY

The present disclosure is made in consideration of the circumstancesdescribed above, and relates to both reducing computational complexityand enabling an evaluation of the reliability of output data obtained byinputting data to a trained model.

An information processing system according to a first aspect includes aninformation processing device and a control device. The informationprocessing device includes: a first determination section that makes adetermination as to whether a change amount of second output datarelative to first output data is not more than a pre-specifiedthreshold, the first output data being obtained from a trained model byinput of first input data to the trained model, the trained model beingtrained beforehand, and the second output data being obtained from thetrained model by input of second input data to the trained model, inwhich second input data a perturbation of a specified perturbationamount is applied to the first input data; and a first output sectionthat outputs information representing the first input data and aperturbation amount for which the change amount is determined by thefirst determination section to be not more than the threshold. Thecontrol device includes: an acquisition section that acquires thirdinput data to be inputted to the trained model; a second determinationsection that makes a determination as to whether the third input data iscontained in a range represented by the information outputted by thefirst output section, the range being the perturbation amount measuredfrom the first input data; and a second output section that outputs thethird input data when the third input data is determined by the seconddetermination section to be not contained in the range. The firstdetermination section makes the determination thereof with the thirdinput data outputted by the second output section serving as the firstinput data, and the first output section outputs the third input dataand a perturbation amount for which the change amount is determined bythe first determination section to be not more than the threshold.

According to the information processing system according to the firstaspect, the trained model has been trained beforehand. The first outputdata is obtained from the trained model by input of the first input datato the trained model. A perturbation of a specified perturbation amountis applied to the first input data to obtain the second input data, andthe second output data is obtained from the trained model by input ofthe second input data to the trained model. The information processingdevice makes a determination as to whether a change amount of the secondoutput data relative to the first output data is at most thepre-specified threshold. The information processing device then outputsinformation representing the first input data and a perturbation amountfor which the change amount is judged to be less than or equal to thethreshold. The control device makes a determination as to whether thethird input data to be inputted to the trained model is in the rangethat is the perturbation amount measured from the first input data,which are represented by the information outputted by the first outputsection. Then, if the third input data is judged to be not contained inthis range, the control device outputs the third input data. Theinformation processing device then carries out the determinationdescribed above, using the third input data outputted by the controldevice as the first input data. When the change amount is determined tobe less than or equal to the threshold, the information processingdevice further outputs information representing the third input data andthe perturbation amount for which this determination is made.

Thus, both computational complexity may be reduced and the reliabilityof output data obtained by inputting data to the trained model may beevaluated.

In an information processing system according to a second aspect, in theinformation processing system according to the first aspect, the firstdetermination section repeats processing until the change amount is notmore than the threshold, the processing including, when the changeamount exceeds the threshold, reducing the perturbation amount from thepreceding value thereof, obtaining new the second output data with thereduced perturbation amount, and making a determination as to whetherthe change amount of the obtained second output data relative to thefirst output data is not more than the threshold.

According to the information processing system according to the secondaspect, because the determination is repeated until the change amount isless than or equal to the threshold, the reliability of output dataobtained by inputting data to the trained model may be evaluatedaccurately.

In an information processing system according to a third aspect, in theinformation processing system according to the first aspect or secondaspect, the first output section represents the first input data as apoint in a space with the same number of dimensions as the first inputdata and outputs, as the information, a hypersphere centered on thepoint whose radius is the perturbation amount.

According to the information processing system according to the thirdaspect, because the first input data and the perturbation amount arerepresented as a hypersphere, the information representing the firstinput data and perturbation amount may be easy to handle.

In an information processing system according to a fourth aspect, in theinformation processing system according to any one of the first to thirdaspects, the information processing device includes a server computer,the control device includes a computer mounted at a vehicle, and thefirst input data, second input data and third input data are image data.

In an information processing system according to a fifth aspect, in theinformation processing system according to any one of the first tofourth aspects, the acquisition section acquires image data imaged by acamera mounted at the vehicle to serve as the first input data.

In an information processing system according to a sixth aspect, in theinformation processing system according to any one of the first to fifthaspects, the information outputted by the first output section is usedin conducting autonomous driving control.

According to various aspects of the present disclosure, bothcomputational complexity may be reduced and the reliability of outputdata obtained by inputting data to a trained model may be evaluated.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram showing an example of structures of aninformation processing system.

FIG. 2 is a block diagram showing an example of hardware structures ofan information processing device.

FIG. 3 is a diagram for describing a trained model.

FIG. 4 is a block diagram showing an example of hardware structures of acontrol device.

FIG. 5 is a block diagram showing an example of functional structures ofthe information processing device and control device.

FIG. 6 is a flowchart showing an example of first output processing.

FIG. 7 is a flowchart showing an example of second output processing.

DETAILED DESCRIPTION

Herebelow, an exemplary embodiment of the present disclosure isdescribed in detail with reference to the attached drawings.

First, structures of an information processing system 10 according tothe present exemplary embodiment are described with reference to FIG. 1. As shown in FIG. 1 , the information processing system 10 includes aninformation processing device 12 and a control device 14. Theinformation processing device 12 is employed in a testing phase. Aserver computer or the like can be mentioned as an example of theinformation processing device 12. The control device 14 is provided in avehicle and is employed in an operation phase. An on-board computer suchas an electronic control unit (ECU) or the like can be mentioned as anexample of the control device 14.

Now, hardware structures of the information processing device 12according to the present exemplary embodiment are described withreference to FIG. 2 . As shown in FIG. 2 , the information processingdevice 12 includes a central processing unit (CPU) 20, a memory 21 thatserves as a temporary storage area, and a nonvolatile memory unit 22.The information processing device 12 further includes a display unit 23such as a liquid crystal display or the like, an input unit 24 such as akeyboard and mouse or the like, and a network interface (I/F) 25 that isconnected to a network. The CPU 20, memory 21, memory unit 22, displayunit 23, input unit 24 and network interface 25 are connected to a bus26.

The memory unit 22 is embodied by a hard disk drive (HDD), a solid statedrive (SSD), a flash memory or the like. An information processingprogram 30 is memorized in the memory unit 22 serving as a memorymedium. The CPU 20 reads the information processing program 30 from thememory unit 22, loads the information processing program 30 into thememory 21, and executes the loaded information processing program 30.

A trained model 32 and range data 34 are memorized in the memory unit22. The trained model 32 is a model that has been trained beforehand bymachine learning. The trained model 32 is used for identifying objectsappearing in images, which are input data. As in the example illustratedin FIG. 3 , the trained model 32 according to the present exemplaryembodiment is a deep neural network model including a single inputlayer, plural intermediate layers, and a single output layer. An imageof an object is inputted to the trained model 32 and, in response tothis input, an identification result for the object appearing in theimage and a probability of the identification result are outputted. Roadsigns, traffic signals, vehicles, people and so forth can be mentionedas examples of objects. Note that the trained model 32 is not limited tobeing a deep neural network model. Details of the range data 34 aredescribed below.

Now, hardware structures of the control device 14 according to thepresent exemplary embodiment are described with reference to FIG. 4 . Asshown in FIG. 4 , the control device 14 includes a CPU 40, a memory 41that serves as a temporary storage area, and a nonvolatile memory unit42. The control device 14 further includes a network interface 43 thatis connected to the network, and an input/output interface 44. Anon-board camera 46 is connected to the input/output interface 44. TheCPU 40, memory 41, memory unit 42, network interface 43 and input/outputinterface 44 are connected to a bus 45.

The memory unit 42 is embodied by an HDD, SSD, flash memory or the like.A control program 50 is memorized in the memory unit 42 serving as amemory medium. The CPU 40 reads the control program 50 from the memoryunit 42, loads the control program 50 into the memory 41, and executesthe loaded control program 50. Similarly to the memory unit 22, thetrained model 32 and the range data 34 are also memorized in the memoryunit 42.

The on-board camera 46 is mounted in a cabin of the vehicle, imagesforward of the vehicle, and outputs obtained images to the controldevice 14.

Now, functional structures of the information processing device 12 andcontrol device 14 according to the present exemplary embodiment aredescribed with reference to FIG. 5 . As shown in FIG. 5 , theinformation processing device 12 includes a first acquisition section60, a derivation section 62, a first determination section 64 and afirst output section 66. By executing the information processing program30, the CPU 20 of the information processing device 12 functions as thefirst acquisition section 60, the derivation section 62, the firstdetermination section 64 and the first output section 66. The controldevice 14 includes a second acquisition section 70, a seconddetermination section 72 and a second output section 74. By executingthe control program 50, the CPU 40 of the control device 14 functions asthe second acquisition section 70, the second determination section 72and the second output section 74.

The first acquisition section 60 acquires test data y_t. The test datay_t according to the present exemplary embodiment is image data imagedby an imaging device, such as an on-board camera or the like, in whichan object appears. The test data y_t corresponds to the first input datarelating to the technology of the disclosure.

The derivation section 62 derives a perturbation amount ε of aperturbation to be applied to the test data y_t. More specifically, thederivation section 62 uses a technique recited in the below-mentionedReference Document 1 to derive a perturbation amount s that is a verysmall (for example a minimum) amount among perturbations to the testdata y_t that cause changes with change amounts exceeding a thresholdTH, which are described below. Examples of the perturbation referred toherein that can be mentioned include: deciding a maximum value forchange amounts of several pixels in an image and changing pixel values;deciding a maximum value for an absolute value of an average of pixelvalues or a variance value of pixel values and applying white noise toan image; and so forth. In many cases, the maximum value corresponds tothe perturbation amount ε herein.

Reference Document 1: Szegedy, Zaremba, Sutskever, Bruna, Ethan,Goodfellow and Fergus, Intriguing properties of neural networks,(ICLR2013).

The first determination section 64 inputs the test data y_t to thetrained model 32 and hence obtains first output data from the trainedmodel 32. The first determination section 64 further inputs test datay_tε, in which a perturbation of the perturbation amount ε is applied tothe test data y_t, to the trained model 32 and hence obtains secondoutput data from the trained model 32. The test data y_tε corresponds tothe second input data relating to the technology of the disclosure.

The first determination section 64 derives a change amount C of thesecond output data relative to the first output data. In the presentexemplary embodiment, an example is described in which an absolute valueof the difference between a probability outputted from the trained model32 as the first output data and a probability outputted from the trainedmodel 32 as the second output data is employed as the change amount C.The first determination section 64 makes a determination as to whetherthe derived change amount C is not more than the pre-specified thresholdTH. The first determination section 64 repeats this processing—reducingthe perturbation amount ε from the preceding value thereof, obtainingsecond output data with the reduced perturbation amount ε, anddetermining whether the change amount C of the newly obtained secondoutput data relative to the first output data is not more than thethreshold TH—until the change amount C is not more than the thresholdTH. How much the perturbation amount cis reduced in this configurationmay be specified beforehand and may be, for example, reduction by apre-specified proportion (for example, 1%) of the preceding value of theperturbation amount ε.

The first output section 66 outputs (saves) information representing thetest data y_t and perturbation amount ε, for which the change amount Chas been determined by the first determination section 64 to be not morethan the threshold TH, to the memory unit 22 to serve as the range data34. In the present exemplary embodiment, the first output section 66outputs a hypersphere to the memory unit 22 as the range data 34, inwhich the test data y_t is represented as a point in a space with thesame number of dimensions as the test data y_t and the hypersphere iscentered on that point with a radius of the perturbation amount ε. Forexample, if the test data y_t is a monochrome image (that is, an imagein which each pixel has one of two values, 0 or 1) and the resolution ofthe image is 10 pixels by 10 pixels, the test data y_t may berepresented as a point in space with 2×10×10 dimensions. The hyperspherecan be considered to represent images for which change amounts of theoutput of the trained model 32 are small when perturbation amounts ε areapplied.

The information processing device 12 carries out the processingdescribed above on each of plural sets of different test data y_t. Thus,the range data 34 forms a union of plural hyperspheres obtained for therespective sets of different test data y_t. The information processingdevice 12 may shorten a computation duration by carrying out theprocessing described above for each of a plural number of sets ofdifferent test data y_t in parallel, with a pre-specified degree ofparallelism. The degree of parallelism that is employed in thissituation may be, for example, a number of cores of the CPU 20.

The range data 34 memorized in the memory unit 22 by the processingdescribed above is sent from the information processing device 12 to thecontrol device 14 and is memorized in the memory unit 42 of the controldevice 14. The first output section 66 may output (send) the range data34 to the control device 14 via the network interface 25.

The second acquisition section 70 acquires image data imaged by theon-board camera 46 via the input/output interface 44. This image data isimaged by the on-board camera 46 at a pre-specified frame rate, forexample, while an ignition switch of the vehicle in which the controldevice 14 is mounted is turned on. Below, this image data is referred toas actual data y_h. The actual data y_h corresponds to the third inputdata, to be inputted to the trained model 32, relating to the technologyof the disclosure. The second acquisition section 70 corresponds to theacquisition section relating to the technology of the disclosure. In thepresent exemplary embodiment, the test data y_t and the actual data y_hhave the same number of dimensions (that is, the numbers of bits inpixels and the number of pixels). If, however, the test data y_t and theactual data y_h have different numbers of dimensions, image processingmay be applied to one or both of the test data y_t and the actual datay_h such that they have the same number of dimensions.

The second determination section 72 makes a determination as to whetherthe actual data y_h is contained in a range R of the perturbation amountε measured from the test data y_t, which are represented by theinformation outputted by the first output section 66. More specifically,when the actual data y_h is represented as a point with the same numberof dimensions as the actual data y_h, the second determination section72 makes a determination as to whether this point is contained in theunion of plural hyperspheres represented by the range data 34.

If the second determination section 72 determines that the actual datay_h is contained in the range R, the second output section 74 inputs theactual data y_h to the trained model 32. In response to this input, anidentification result of an object included in the actual data y_h and aprobability of the identification result are outputted from the trainedmodel 32. Outputs from the trained model 32 are used in, for example,autonomous driving control of the vehicle.

On the other hand, if the second determination section 72 determinesthat the actual data y_h is not contained in the range R, the secondoutput section 74 outputs (sends) the actual data y_h to the informationprocessing device 12 via the network interface 43. The informationprocessing device 12 treats the actual data y_h sent from the controldevice 14 as test data y_t. That is, the first determination section 64conducts the determinations described above, using the actual data y_houtputted by the second output section 74 as the test data y_t. Then,the first output section 66 outputs information representing the actualdata y_h and a perturbation amount ε, for which the change amount C hasbeen determined by the first determination section 64 to be not morethan the threshold TH, to the memory unit 22 as the range data 34. Thus,actual data y_h that is not contained in the range R is fed back fromthe control device 14 to the information processing device 12, and therange data 34 is updated on the basis of the fed back actual data y_h.Therefore, the reliability of output data that is obtained by inputtingdata to the trained model 32 may be evaluated.

The second output section 74 may output (save) actual data y_h that isnot contained in the range R to the memory unit 42. In this case, thesecond output section 74 outputs the actual data y_h memorized in thememory unit 22 to the information processing device 12 with apre-specified timing. For example, a periodic timing, times at which anumber of sets of the actual data y_h memorized in the memory unit 42exceeds a certain number, and the like can be mentioned as examples ofthe pre-specified timing. Further, when the second determination section72 determines that actual data y_h is not contained in the range R, thesecond output section 74 may give a warning by one or both of a voicemessage and an indication on a display.

Now, operation of the information processing device 12 according to thepresent exemplary embodiment is described with reference to FIG. 6 .First output processing, which is depicted in FIG. 6 , is implemented bythe CPU 20 of the information processing device 12 executing theinformation processing program 30. The first output processing isexecuted, for example, when an execution command is inputted by a uservia the input unit 24. The first output processing is executed inparallel for each of a plural number of sets of test data y_t.

In step S10 of FIG. 6 , the first acquisition section 60 acquires testdata y_t. In step S12, in the manner described above, the derivationsection 62 derives a perturbation amount ε of a perturbation to beapplied to the test data y_t acquired in step S10.

In step S14, the first determination section 64 inputs the test data y_tacquired in step S10 to the trained model 32 and hence acquires firstoutput data from the trained model 32. Further, the first determinationsection 64 inputs test data y_tε, in which a perturbation of theperturbation amount ε is applied to the test data y_t acquired in stepS10, to the trained model 32 and hence acquires second output data fromthe trained model 32. When step S14 is initially executed, theperturbation amount ε is the perturbation amount ε derived in step S12.When step S14 is executed a second or subsequent time after step S18,which is described below, the perturbation amount ε is the perturbationamount ε that has been reduced in step S18. In the manner describedabove, the first determination section 64 derives a change amount C ofthe second output data relative to the first output data.

In step S16, the first determination section 64 makes a determination asto whether the derived change amount C is not more than thepre-specified threshold TH. If the result of this determination isnegative, the processing advances to step S18. In step S18, the firstdetermination section 64 reduces the perturbation amount ε to a valuesmaller than the value used in the preceding step S14. When theprocessing of step S18 is complete, the processing returns to step S14.

Alternatively, when the result of the determination in step S16 isaffirmative, the processing advances to step S20. In step S20, the firstoutput section 66 outputs information representing the test data y_t andperturbation amount ε, for which it has been determined in step S16 thatthe change amount C is not more than the threshold TH, to the memoryunit 22 to serve as the range data 34. When the processing of step S20is complete, the first output processing ends.

Now, operation of the control device 14 according to the presentexemplary embodiment is described with reference to FIG. 7 . Secondoutput processing, which is depicted in FIG. 7 , is implemented by theCPU 40 of the control device 14 executing the control program 50. Thesecond output processing is executed, for example, each time image dataimaged by the on-board camera 46 (actual data y_h) is inputted to thecontrol device 14.

In step S30 of FIG. 7 , the second acquisition section 70 acquires theactual data y_h imaged by the on-board camera 46 via the input/outputinterface 44. In step S32, in the manner described above, the seconddetermination section 72 makes a determination as to whether the actualdata y_h acquired in step S30 is contained in the range R of theperturbation amount ε measured from the test data y_t represented by theinformation outputted in step S20. If the result of this determinationis affirmative, the processing advances to step S34, and if the resultof the determination is negative, the processing advances to step S36.

In step S34, the second output section 74 inputs the actual data y_hacquired in step S30 to the trained model 32. In response to this input,an identification result of an object included in the actual data y_hand a probability of the identification result are outputted from thetrained model 32. Outputs from the trained model 32 are used in, forexample, autonomous driving control of the vehicle. When the processingof step S34 is complete, the second output processing ends.

In step S36, in the manner described above, the second output section 74outputs the actual data y_h acquired in step S30 to the informationprocessing device 12 via the network interface 43. When the processingof step S36 is complete, the second output processing ends.

As described above, according to the present exemplary embodiment, bothcomputational complexity may be reduced and the reliability of outputdata obtained by inputting data to a trained model may be evaluated.

The derivation section 62 and the first determination section 64 of theexemplary embodiment described above may be configured as a singlefunctional section. In this configuration, it is acceptable for thisfunctional section to use, for example, the technique recited in thefollowing Reference Document 2 to derive a maximum perturbation amount εfor which the change amount C is not more than the threshold TH.

Reference Document 2: Wong and Kolter, Provable defenses againstadversarial examples via the convex outer adversarial polytope,(ICML2018).

The processing that is implemented by the CPU 20 and CPU 40 of theexemplary embodiment described above is described as software processingthat is implemented by executing programs. However, each processing maybe processing that is implemented by hardware such as an ASIC(Application-Specific Integrated Circuit), an FPGA (Field-ProgrammableGate Array) or the like. Further, the processing executed by the CPU 20and the CPU 40 may be processing that is implemented by a combination ofboth software and hardware. The information processing program 30memorized in the memory unit 22 and the control program 50 memorized inthe memory unit 42 may be recorded and distributed on various recordingmedia.

The present disclosure is not limited by the above recitations. Inaddition to the above recitations, it will be clear that numerousmodifications may be embodied within a technical scope not departingfrom the gist of the disclosure.

What is claimed is:
 1. An information processing system comprising aninformation processing device and a control device, the informationprocessing device including: a first processor that is configured tomake a determination as to whether each of a plurality of change amountscorresponding to each of a plurality of second output data relative toeach of a plurality of first output data is not more than apre-specified threshold, the plurality of first output data beingobtained from a trained model by input of a plurality of first inputdata to the trained model, the plurality of first input data beingrepresented as a first set of points in a space with the same number ofdimensions as the first input data, and the trained model being trainedbeforehand, the plurality of second output data being obtained from thetrained model by input of a plurality of second input data to thetrained model, the plurality of second input data being represented as asecond set of points in the space with the same number of dimensions asthe second input data, and each second input data is obtained byapplying a perturbation of a specified perturbation amount, for whichthe corresponding change amount is determined to be not more than thethreshold, to the corresponding first input data such that the eachsecond data is contained in a hypersphere whose radius is the specifiedperturbation amount and which is centered at a point representing thecorresponding first input data; obtaining a union of a plurality ofhyperspheres each of which corresponds to the hypersphere that containsthe each second data; output the union of the plurality of hyperspheresas a range information, and the control device including: a secondprocessor that is configured to acquire third input data to be inputtedto the trained model; make a determination as to whether the third inputdata is contained in the union of the plurality of hyperspheresrepresented by the range information outputted by the informationprocessing device; and output the third input data when the third inputdata is determined to be not contained in the union of the plurality ofhyperspheres, wherein the first processor is configured to receive thethird input data, which is determined to be not contained in the unionof the plurality of hyperspheres, and make the determination thereofwith the third input data being added to the plurality first input datasuch that the range information is updated in correspondence with thethird input data being added to the plurality of the first data, and thefirst processor is configured to output the third input data and aperturbation amount for which the corresponding change amount isdetermined to be not more than the threshold.
 2. The informationprocessing system according to claim 1, wherein the first processor isconfigured to repeat processing until each of the plurality of changeamounts is not more than the threshold, the processing including, when achange amount of the plurality of change amounts exceeds the threshold,reducing the perturbation amount from the preceding value thereof,obtaining a new second output data with the reduced perturbation amount,and making a determination as to whether a new change amount of theobtained new second output data relative to a first output data is notmore than the threshold.
 3. The information processing system accordingto claim 2, wherein the information processing device includes a servercomputer, the control device includes a computer mounted at a vehicle,and the plurality of first input data, the plurality of second inputdata and the third input data are image data.
 4. The informationprocessing system according to claim 2, wherein the second processor isconfigured to acquire image data imaged by a camera mounted at thevehicle to serve as the plurality of first input data.
 5. Theinformation processing system according to claim 2, wherein theinformation outputted by the first processor is used in conductingautonomous driving control.
 6. The information processing systemaccording to claim 1, wherein the information processing device includesa server computer, the control device includes a computer mounted at avehicle, and the plurality of first input data, the plurality of secondinput data and the third input data are image data.
 7. The informationprocessing system according to claim 1, wherein the second processor isconfigured to acquire image data imaged by a camera mounted at thevehicle to serve as the plurality of first input data.
 8. Theinformation processing system according to claim 1, wherein theinformation outputted by the first processor is used in conductingautonomous driving control.
 9. The information processing systemaccording to claim 1, wherein the plurality of first input data and theplurality of second input data are image data with pixels representingcoordinates of a point in the space with a number of dimensions equal tothe number of pixels in the image data.
 10. The information processingsystem according to claim 1, wherein the determination made by the firstprocessor with respect to the plurality of first output data and theplurality of second output data is performed in parallel, with apre-specified degree of parallelism.
 11. The information processingsystem according to claim 1, wherein the second processor is configuredto store the third input data when the third input data is determined tobe not contained in the union of the plurality of hyperspheres in amemory and output the third input data at a specified timing.